Question 15.23: The voltage transfer function for a certain highpass circuit...
The voltage transfer function for a certain highpass circuit is
H_ν=\frac{j f / f_0}{1+j f / f_0}
Use the maximum passband gain as the reference gain and find the half-power width of the stopband.
The squared voltage gain is given by
A_ν{ }^2=\frac{\left(f / f_0\right)^2}{1+\left(f / f_0\right)^2}
The maximum gain occurs for f \rightarrow \infty , where A_ν(f) \rightarrow 1, so the squared reference gain is A_0^2=A_ν{ }^2(\infty)=1. The half-power width of the stopband consists of the frequencies for which
\begin{aligned} \frac{A^2}{A_0^2} \leq \frac{1}{2} & \Rightarrow \frac{\left(f / f_0\right)^2}{1+\left(f / f_0\right)^2} \leq \frac{1}{2} \Rightarrow \frac{f}{f_0} \leq 1 \\\\ & \Rightarrow 0 \leq f \leq f_0 \end{aligned}
The half-power width of the stopband is f_0 .